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For GLMs in the exponential family, we can obtain the standard errors for the regression coefficients as a function of the diagonal of the fisher information matrix. Does this still hold if the regression distribution is not in the exponential family (this is of course technically not a GLM but I'm not sure if there is a technical name for this kind of model)? For example, beta-binomial or dirchlet-multinomial? In this case, does it instead become necessary to use the diagonal of the Hessian?

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